27
Views
3
CrossRef citations to date
0
Altmetric
Original Articles

Analysis of preconditioners for conjugate gradients through distribution of eigenvalues

&
Pages 135-158 | Received 20 Mar 1995, Published online: 19 Mar 2007

References

  • Axelsson , O. 1972 . A generalized SSOR method . BIT , 12 ( 4 ) : 443 – 467 .
  • Axelsson , O. 1974 . On the efficiency of a class of A-stable methods . BIT , 14 ( 3 ) : 279 – 287 .
  • Beawens , R. 1989 . Proceedings of Preconditioned Conjugate Gradient Methods . Modified incomplete factorization strategies . 1989 . pp. 1 – 16 . Berlin : Springer-Verlag . In Lecture Notes in Mathematics 1457
  • Campos, filho F. F. Rollett J. S Tools for analysis of convergence of Conjugate Gradient-type Methods OUCL Report NA92/23 Oxford University Computing Laboratory Oxford 1992
  • Engeli , M. , Ginsburg , Th. , Rutishauser , H. and Stiefel , E. 1992 . Refined iterative methods for computation of the solution and the eigenvalues of self-adjoint boundary value problems , Stuttgard : Birkhauser Verlag .
  • Engeli , M. , Ginsburg , Th. , Rutishauser , H. and Stiefel , E. 1959 . Refined iterative methods for computation of the solution and the eigenvalues of self-adjoint boundary value problems , Stuttgard : Birkhauser Verlag .
  • Evans , D. J. 1968 . The use of pre-conditioning in iterative methods for solving linear equations with symmetric positive definite matrices . J. Inst. Maths. Applics. , 4 ( 3 ) : 295 – 314 .
  • Evans , D. J. 1973 . “ The analysis and application of sparse matrix algorithms in the finite element method ” . In The Mathematics of Finite Elements and Applications , Edited by: Whiteman , J. R. 427 – 447 . New York : Academic Press .
  • Forsythe , G. E. and Straus , E. G. 1955 . On best conditioned matrices . Proceedings of the Amer. Math. Soc. , 6 : 340 – 345 . Amer. Math. Soc
  • Frankel , S. P. 1950 . Convergence rates of iterative treatments of partial differential equations . MTAC , 4 ( 30 ) : 65 – 75 .
  • Gill , P. E. and Murray , W. 1974 . Newton-type methods for unconstrained and linearly constrained optimization . Math. Prog. , 7 ( 3 ) : 311 – 350 .
  • Gill , P. E. , Murray , W. and Wright , M. H. 1981 . Practical Optimization , Academic Press .
  • Golub , G. H. and O′ Leary , D. P. 1989 . Some history of the conjugate gradient and Lanczos algorithms: 1948-1976 . SIAM Review , 31 ( 1 ) : 50 – 102 .
  • Golub , G. H. and Van Loan , C. F. 1989 . Matrix Computations , The Johns Hopkins University Press .
  • Gustafsson , I. 1978 . A class of first order factorization methods . BIT , 18 ( 2 ) : 142 – 156 .
  • Gustafsson , I. 1989 . Proceedings of Preconditioned Conjugate Gradient Methods . A class of preconditioned conjugate gradient methods applied to finite element equations . 1989 . pp. 44 – 57 . Berlin : Springer-Verlag . In Lecture Notes in Mathematics 1457
  • Hestenes , M. R. 1956 . “ The conjugate-gradient method for solving linear systems ” . In Proceedings of the Sixth Symposium in Applied Mathematics , 83 – 102 . New York : McGraw-Hill . 1953
  • Hestenes , M. R. and Stiefel , E. 1952 . Methods of conjugate gradients for solving linear systems . J. Res. Nat. Bur. Standards , 49 ( 6 ) : 409 – 436 .
  • Jennings , A. 1977 . Influence of the eigenvalue spectrum on the convergence rate of the conjugate gradient method . J. Inst. Maths. Applics. , 20 ( 1 ) : 61 – 72 .
  • Kershaw , D. S. 1978 . The incomplete Cholesky-conjugate gradient method for the iterative solution of systems of linear equations . J. Comp. Phys. , 26 ( 1 ) : 43 – 65 .
  • Meijerink , J. A. and van der Vorst , H. A. 1977 . An iterative solution method for linear system of which the coefficient matrix is a symmetric M-matrix . Math. Comp. , 31 ( 137 ) : 148 – 162 .
  • Reid , J. K. 1970 . Proceedings of the Oxford conference on the Institute of Mathematics and Its Applications held . On the method of conjugate gradients for the solution of large sparse systems of linear equations . April 1970 . pp. 231 – 254 . London : Academic Press . In Large Sparse Sets of Linear Equations
  • Schlick , T. 1993 . Modified Cholesky factorizations for sparse preconditioners . SIAM J. Sci. Comput. , 14 ( 2 ) : 424 – 445 .
  • Sheldon , J. W. 1955 . On the numerical solution of elliptic difference equations . MTAC , 9 ( 51 ) : 101 – 112 .
  • Stone , H. L. 1968 . Iterative solution of implicit approximations of multidimensional partial differential equations . SIAM J. Numer. Anal. , 5 ( 3 ) : 530 – 558 .
  • Turing , A. M. 1948 . Rounding-off errors in matrix processes . Quart. J. of Mech. and Appl. Math. , 1 ( 3 ) : 287 – 308 .
  • van der Vorst , H. A. 1989 . Proceedings of Preconditioned Conjugate Gradient Methods . The convergence behaviour of preconditioned CG and CG-S in the presence of rounding errors . 1989 . pp. 126 – 136 . Berlin : Springer-Verlag . In Lecture Notes in Mathematics 1457
  • Wachspress , E. L. 1963 . Extended application of alternating direction implicit iteration model problem theory . J. Soc. Industr. Appl. Math. , 11 ( 4 ) : 994 – 1016 .
  • Wesson P.J. Parallel algorithms for systems of equations PhD thesis Oxford University Computing Laboratory 1992
  • Wong Y. S. Iterative methods for problems in Numerical Analysis PhD thesis Oxford University Computing Laboratory 1978
  • Young D. M. Iterative methods for solving partial difference equations of elliptic type PhD thesis Harvard University 1950
  • Young , D. M. 1954 . Iterative methods for solving partial difference equations of elliptic type. Amer. Math. Soc., Trans. , 76 ( 4 ) : 92 – 111 .

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.